Task 1.2: Fully Coupled Hydrogeophysical Inversion of Salt-Tracer Experiments RECORD PhD Retreat 9 th -10 th June 2009 Davina Pollock, Center for Applied

Task 1.2: Fully Coupled Hydrogeophysical Inversion of Salt-Tracer Experiments

RECORD PhD Retreat9th-10th June 2009

Davina Pollock, Center for Applied Geoscience, University of Tübingen / ETH Zürich

Zentrum für Angewandte Geowissenschaften (ZAG)Hydrogeologie

Introduction Hydraulic conductivity (K) = key groundwater parameter Measurements:

Wells expensive low resolution Geophysics does not measure directly hydraulic conductivity

petrophysical relationships

Use geophysics to monitor hydraulic testsRelate change in geophysical properties to hydraulic stress Electrical Resistivity Tomography (ERT) during salt-tracer

tests Various previous studies

(e.g. Binley et al., 1996; Slater et al., 2000; Binley et al., 2002; Kemna et al., 2002; Vanderborght et al., 2005; Singha and Gorelick, 2005, Cassiani, 2006)


General Approach Standard: classical geophysical inversion followed by

hydraulic interpretation of concentration distribution decoupled risk of non-physical results (e.g., loss of mass) no direct inference of hydraulic conductivity

Approach in which hydraulics and geophysics are considered as a coupled system:

physics of groundwater flow, transport, and geoelectrics satisfied by definition

efficient and stable inversion strategy?


Our Approach

Time-lapse ERT during tracer test Analyse temporal moments of electrical-potential

perturbations Relate mean arrival time of electrical signal to hydraulic

conductivity Temporal moment generating equations Incorporation into geostatistical inversion


Temporal Moments of Concentration Definition of zeroth and first moments:

Advection Dispersion Equation:

Moment generating equation:

Already used in inversion(e.g. Harvey and Gorelick, 1995; Cirpka and Kitanidis 2000; James et al., 2000; Nowak and Cirpka, 2006)

0 cctc Dq

ck

ck

ck mkmm 1 Dq

01

00

,

,

dtttcm

dttcm

c

c

xx

xx


Temporal Moments of Potential Perturbations Definition

Poisson equation

Initial distribution φ0

Linearisation perturbation φ‘

Moment generating equation

ioIc xxxx 0

00 c

00 ckk mm

ioI xxxx 00

0

, dtttm kk xx

(Pollock & Cirpka, 2008)

Valid if κc<<σ0 and for m1/m0


Setup Quasi two-dimensional sandbox Injection of salt tracer ERT monitoring using different electrode configurations Results shown are for a hypothetical test case

Electrodes

Fixed head

Fixed head



Measurements

tc=m1/m0

m0


Geostatistical Inversion Geostatistical regularisation (Kitanidis, 1995)

parameters considered spatially correlated random variables deterministic trend Xβ plus random deviation Y′: maximise posterior likelihood of parameters minimise objective function:

Penalise not meeting the measurementsDeviation from prior mean valuePenalise spatial fluctuations

'1

θ|YY'*1

ββ*

'1ZZ

''

YRYββRββ

β,YZZRβ,YZZθ,Z|β,YT

''

)(T

mTmmL

XβYY '


depends onelectrode configuration


Final Remarks Encouraging results:

larger features in ln(K) resolved physical behaviour guaranteed

Not all details resolved in ln(K) field but also in time curves of electrical signals

include higher order moments as data? Starting experiments in 2-D sandboxapplication to real lab data Necessary extensions for field applications:

3-D different boundary conditions

Documents

Task 1.2: Fully Coupled Hydrogeophysical Inversion of Salt-Tracer Experiments RECORD PhD Retreat 9 th -10 th June 2009 Davina Pollock, Center for Applied